In Δ ABC, AD is the bisector of $\angle A$, meeting side BC at D. If AB = 10 cm, AC = 14 cm, and BC = 6 cm, find the lengths of segments BD and D

Step 1: Understand the Angle Bisector Theorem

The Angle Bisector Theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. In $\triangle ABC$, AD bisects $\angle A$, so it divides side BC into segments BD and DC, such that the ratio of AB to AC is equal to the ratio of BD to DC.

Step 2: Substitute given values into the theorem

We are given the lengths of the sides: AB = 10 cm and AC = 14 cm. We substitute these values into the Angle Bisector Theorem equation.

Step 3: Simplify the ratio and express BD in terms of DC

We simplify the ratio $\frac{10}{14}$ to $\frac{5}{7}$. Then, we can express BD in terms of DC, which will be useful for the next step.

Step 4: Use the total length of BC

We know that the total length of side BC is 6 cm, and it is composed of segments BD and DC. So, $BC = BD + DC$. We substitute the given value of BC and the expression for BD from the previous step.

Step 5: Solve for DC

Combine the terms involving DC. To solve for DC, multiply both sides by the reciprocal of $\frac{12}{7}$. Simplify the fraction to get the length of DC.

Step 6: Solve for BD

Since $BC = BD + DC$, we can find BD by subtracting DC from BC. Substitute the known values to find the length of BD.

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